Short version: Start with a closed interval of length $t>0$ and repeatedly remove a random and uniformly distributed subinterval of length $1$ so long as this is possible. For $t$ large, what is the asymptotic of the expected value $\varphi(t)$ of the number of steps that this process will run? ...

For an algebraic number $\alpha$, let $f_\alpha$ denote its minimal polynomial. We can symbolically represent an algebraic number $\alpha$ by the tuple $$ (f_\alpha, x, y, r) \in \mathbb{Q}[x] \times \mathbb{Q}^3 $$ so that $\alpha$ is the unique root of $f_\alpha$ inside the circle centered at ...

If we restrict all parameters in the set building axioms of $\sf ZFC$ to definable sets, would the celebrated Cantor's theorem still apply? Can existence of uncountable sets be proven at all? The set building axioms of $\sf ZFC$ are the axioms of: pairing, union, powerset, Separation and Replacem...

Let $(X,D_X)$ and $(S, D_S)$ be smooth normal crossings pairs over $\mathbb C$; i.e. smooth schemes of finite type over $\mathbb C$ with a normal crossings divisor. If $f:X \to S$ is a proper, flat and log-smooth (with respect to the log structures given by the divisors) such that $f^{-1}(D_S)=D_...